Disordered plasmonic nanostructures, known for their broadband absorption, can transition into tunable narrowband reflectors when coupled to a lossless optical cavity. This effect arises from cavity-induced suppression of intrinsic decay in specific optical modes, causing those modes to reflect rather than be absorbed. By varying the spacer thickness, the system can be tuned across the visible spectrum, enabling precise control over optical response. This mechanism offers a simple yet powerful approach to structural color generation, producing vivid and reconfigurable colors without relying on periodic nanostructures or complex fabrication.

This notebook reproduces the key results of this phenomenon based on the work Mao, P., Liu, C., Song, F. et al. Manipulating disordered plasmonic systems by external cavity with transition from broadband absorption to reconfigurable reflection. Nat Commun 11, 1538 (2020). DOI: 10.1038/s41467-020-15349-y. The system consists of silver nanoparticles on a LiF spacer layer and a bottom silver mirror. The level of disorder and the spacer thickness can greatly affect the spectral response and color of the system. We will also demonstrate how to convert the reflection spectra to chromaticity coordinates and plot them on a CIE 1913 chromaticity diagram.

Simulation Setup¶

To characterize the color of the disordered plasmonic system, we will simulate the entire visible wavelength range from 380 nm to 780 nm.

The plasmonic nanoparticles and bottom mirror are made of silver. We will use the silver medium from our material library. LiF is characterized by the Sellmeier model with the parameter values extracted from the reference.

In the ordered system, the plasmonic nanoparticles have a diameter d0 of 10 nm. The particles are arranged in a square lattice with an inter-particle spacing of 2 nm. We will investigate an array of 7 by 7 particles. A larger array can be simulated but similar results will be obtained.

Disorder is added to the system by adding random variations to the particle diameters and positions. The level of disorder is controlled by the parameter alpha, where alpha=0 corresponds to the perfectly ordered system. A larger alpha corresponds to more disorder. Specifically,

$$ d = d_0 \left[1 + \alpha U_1(x, y)\right], $$

$$ x^i = x_0^i + (\alpha \cdot d_0) U_2(x, y), $$

$$ y^i = y_0^i + (\alpha \cdot d_0) U_3(x, y), $$

where $U_1(x, y)$, $U_2(x, y)$, and $U_3(x, y)$ are three independent uniform distributions between -1 and 1.

In this case, we will focus on alpha = 0.5 but other disorder levels can be studied in the same way.

With the diameters and positions of each particle, we can generate the particle geometries.

For excitation, we will add a PlaneWave. We will also add a FluxMonitor to measure the reflection spectrum.

The goal is to investigate the reflection spectrum as a function of both wavelength and the spacer layer thickness, which requires us to perform a parameter sweep on different spacer thicknesses. To do so, we define a function make_sim that takes the spacer thickness as the input argument and returns a Simulation object.

Usually we recommend leaving at least half a wavelength of spacing between any structures to the PML boundaries to avoid evanescent field from leaking into PML and causing the simulation to diverge. In this case, we recall that the skin depth of silver at the visible range is in the order of 10 nm. Therefore, including more than half a wavelength of silver mirror thickness in the simulation domain is a waste of computation and cost. Instead, we only need to leave about 100 nm of silver mirror.

Since the nanoparticles can be small, we will need to use a fine grid to resolve them. We use a MeshOverrideStructure around the nanoparticle region so the grid size is 2 nm. Outside of the nanoparticle region, we can use a coarse grid to reduce the simulation cost.

To ensure the simulation setup is correct, create a sample simulation and visualize it. We confirm that the simulation setup looks correct.

As expected, we get a warning about structures being to close to the PML boundary even though in this case we don't worry about it.

Parameter Sweep¶

Now we are ready to perform the parameter sweep. We will sweep the spacer layer thickness from 50 nm to 300 nm.

To avoid seeing many warnings from each simulation, we set the logging level to Error.

Result Visualization¶

After the parameter sweep is done, we can extract the reflectance from each simulation and plot them in a 2D false color map. The result reproduces Fig. 2f of the referenced paper. Here we see two reflectance peaks corresponding to the first and second order modes.

Finally, we will calculate the chromaticity coordinates for each spacer layer thickness. One can calculate chromaticity coordinates (CIE 1931 ($x$, $y$)) from a reflection spectrum by simulating how that reflected light would appear under a given illuminant (e.g., D65) and observer color matching functions:

$$ X = \int R(\lambda) S(\lambda) \overline{x}(\lambda) \, d\lambda, $$ $$ Y = \int R(\lambda) S(\lambda) \overline{y}(\lambda) \, d\lambda, $$ $$ Z = \int R(\lambda) S(\lambda) \overline{z}(\lambda) \, d\lambda, $$

$$ x = \frac{X}{X + Y + Z}, \quad y = \frac{Y}{X + Y + Z}, $$

where $ R(\lambda)$ is the reflectance spectrum, $S(\lambda)$ is the spectral power distribution of the illuminant, $\overline{x}(\lambda)$, $ \overline{y}(\lambda)$, $ \overline{z}(\lambda)$ are the color matching functions. This calculation can be done conveniently by using the Python library color, which can be installed by pip install colour-science.

Finally, we plot the CIE 1931 chromaticity diagram with the sRGB color gamut as a reference. The chromaticity of the disordered plasmonic system is plotted as black dots.

Similarly, we can calculate the sRGB values from each spectrum and see how the color actually looks like.